blssex - Intuitionistic Logic Explorer

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Theorem blssex 14609

Description: Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

**Assertion**
Ref	Expression
blssex	$/\$ $/\$

**Proof of Theorem blssex**
Step	Hyp	Ref	Expression
1		blss 14607	. . . . . . 7 $/\$ $/\$
2		sstr 3188	. . . . . . . . 9 $/\$
3	2	expcom 116	. . . . . . . 8
4	3	reximdv 2595	. . . . . . 7
5	1, 4	syl5com 29	. . . . . 6 $/\$ $/\$
6	5	3expa 1205	. . . . 5 $/\$ $/\$
7	6	expimpd 363	. . . 4 $/\$ $/\$
8	7	adantlr 477	. . 3 $/\$ $/\$ $/\$
9	8	rexlimdva 2611	. 2 $/\$ $/\$
10		simpll 527	. . . . 5 $/\$ $/\$ $/\$
11		simplr 528	. . . . 5 $/\$ $/\$ $/\$
12		rpxr 9730	. . . . . 6
13	12	ad2antrl 490	. . . . 5 $/\$ $/\$ $/\$
14		blelrn 14599	. . . . 5 $/\$ $/\$
15	10, 11, 13, 14	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$
16		simprl 529	. . . . 5 $/\$ $/\$ $/\$
17		blcntr 14595	. . . . 5 $/\$ $/\$
18	10, 11, 16, 17	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$
19		simprr 531	. . . 4 $/\$ $/\$ $/\$
20		eleq2 2257	. . . . . 6
21		sseq1 3203	. . . . . 6
22	20, 21	anbi12d 473	. . . . 5 $/\$ $/\$
23	22	rspcev 2865	. . . 4 $/\$ $/\$ $/\$
24	15, 18, 19, 23	syl12anc 1247	. . 3 $/\$ $/\$ $/\$ $/\$
25	24	rexlimdvaa 2612	. 2 $/\$ $/\$
26	9, 25	impbid 129	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1364

wcel 2164

wrex 2473

wss 3154

crn 4661

cfv 5255 (class class class)co 5919

cxr 8055

crp 9722

cxmet 14035

cbl 14037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-po 4328 df-iso 4329 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-map 6706 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-xneg 9841 df-xadd 9842 df-psmet 14042 df-xmet 14043 df-bl 14045

This theorem is referenced by: blbas 14612 elmopn2 14628 mopni2 14662 metss 14673 tgioo 14733